APPROXIMATE TRANSMISSION NETWORK MODELS
FOR USE IN ANALYSIS AND DESIGN
D. Crevier
Report #MIT-EL 73-008
June 1972
B.O.c-
ENERGY LABORATORY
The Energy Laboratory was established by the Massachusetts Institute
of Technology as a Special Laboratory of the Institute for research on
the complex societal and technological problems of the supply, demand
and consumption of energy.
Its full-time staff assists in focusing
the diverse research at the Institute to permit undertaking of long
term interdisciplinary projects of considerable magnitude.
For any
specific program, the relative roles of the Energy Laboratory, other
special laboratories, academic departments and laboratories depend upon
the technologies and issues involved.
Because close coupling with the nor-
.mal academic teaching and research activities of the Institute is an
important feature of the Energy Laboratory, its principal activities
are conducted on the Institute's Cambridge Campus.
This study was done in association with the Electric Power Systems
Engineering Laboratory and the Department of Civil Engineering (Ralph
M. Parsons Laboratory for Water Resources and Hydrodynamics and the
Civil Engineering Systems Laboratory).
.ra.
2
Table of Contents
Page
. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.
Introduction
2.
"Exact" Model.
3.
Simplified Model I (D-C Load Flow) . . . . . . . .
3.1
. . . . . . . . . . . . . . . . .
5.
6.
· · · ·
7
Analogy between the simplified model and a
resistive network.
4.
· ·
. . . . . .
odel I . .
. . .1 2
3.2
Matrix solution of the Simplified
3.3
Accuracy of Simplified Model
3.4
Uses of Simplified Model I in the literature. . . . . . . 12
3.5
Disadvantages of Simplified Model I . . . . .
I. . . . . . . .
. .. . 14
Simplified Model 2 (Transshipment model) . . . . . . . . . . . 15
4.]
Peculiarities of the transshipment problem and solution
4.2
Analytical formulation of the problem . . . . . . . . .
· 20
4.3
Accuracy of the transshipment algorithm . . . . . . . .
· 23
4.4
Smoothing out procedure . . . . . . . . . . . . . . . .
· 28
4.5
Computing line losses . . . . . . . . . . . . . . . . .
· 30
4.6
Uses of the transshipment algorithm in the literature
· 30
Simplified Model 3 (Ford-Fulkerson model) .
. . . . . . . .
5.1
Advantages and uses of the Ford-Fulkerson algorithm
5.2
Computing line losses .
Conclusion.
.
. . . . . . . . .
........................
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
18
· 33
· 34
· 34
· 36
· 37
.
rm*i
·
lu
3
1.
Introduction
A wide range of power system expansion planning techniques require
a very large number of solutions for the real power flows in the lines
for a given transmission network.
A prime example is of course the
design of the transmission network itself but multiple line flow solutions can also be required in generation expansion planning and in the
associated reliability studies.
The use of complete"A-C load flow"
iteration techniques for solving the exact nonlinear equations can easily
lead to prohibitive computation requirements.
Therefore there is a
definite need for approximate transmission line models and solution
techniques.
It is the purpose of this paper to investigate and compare three
approximate models and solution techniques.
1)
"D-C load Flow Model"
2)
Transshipment Model" - Linear Programming
3)
"Ford-Fulkerson Model"
The models are listed in order of decreasing accuracy and increasing
computational efficiency.
These approximate models are not new.
value of this report lies in the comparison of their natures.
try
to
isolate some of the pitfalls
those methods,
that
We shall
one might encounter in using
and point out in what circumstances they
meaningful results.
The
will
give
4
2.
Exact" Model
It is always possible to represent a three-phase transmission
line by the following so-called a-model:
L
---A
--------
-- V
S
-\
ish
L
A
R
n'
*
V -
AA
rf-
==
t.
-
y
· VYYV
X
u~~~'
v
~
_-
b, 111
~~.
.
l
I,.
sh
Figure 1
The inductance L represents the self-inductance of each phase in
the line, an
the mutual inductance between phases.
The resistance R represents the resistance of all three phases.
The shunt admittances C and Yh
account for the capacitance and
the shunt conductances between the three phases, and each phase and the
ground.
The above model is used by engineers for detailed technical studies,
especially if the line is very long (more than 200 miles).
If the line
is not too long, or if less than perfect accuracy is desired, the shunt
elements can be neglected.
Nbte:
1)
the shunt conductance is usually neglected, even in
very detailed studies;
5
2)
if the line is very long and if the shunt capacitance
becomes too important, it is usually partially
eliminated by installing shunt inductors at both
ends of the line.
For these reasons, we shall consider as our "exact" model of a transmission line the following model, containing only series elements:
PiJ
X
Pi
R
ViBusi
Bus i
Bus
Figure
2
where X = J L.
w
The real power flows in a line connected between buses i and j,
with bus voltages
V A
and VJi
, can be computed as follows:
The complex powers Sij and Sjit
are given by the expression:
J i;
-
sij
SJi
---
j
2
-*2V
=V1
i
Z
V
2Ji Vi V J ( i-6j)
V
2 -V.e
'i
i
e
R
-J
R-JX
(1) tt
(6 -6i)
(2)
R - JX
Qtuantities with a superbar (-) are complex quantities.
For example
Vi = Vi'6ji =Vie
ttAn
asterisk (*) indicates the complex conjugate of a complex quantity.
6
After separating real and imaginary
parts in the above equations, we
get:
1
ij
R2 + X 2
(RV2 - RViV
j cos(
(
2
Pji =2 2 1 22 (
ji
R + X
-RV
RViV
i
- j6j)+XViV
sin(8 - 6j) (3)
i
cos(6i - 6j) -XViVJ
sin(6i
(4)
- 6j)
An expression for the losses in line ij can be obtained by adding
equations
(3) and (4):
PL = Pi
+
PJ
+
R2
L - 112i
RI .+ X2
2V
iV
(5)
cos
cos 8
(5)
If all voltages are inserted in these formulas in line values and
If
kilovolts, then all powers will come out as three-phase megawatts.
the voltages are expressed in the per unit system, the powers will be
in three phase, per unit megawatts.
The reader will appreciate the fact that even for this simplified
model of a transmission line, the equations
describing
the
power flows
are nonlinear, and usually not amenable to an analytic solution.
for
a small
tion of these
computer.
Even
system involving only a few buses, the only method of soluequations is a recursive
solution
algorithm
on a digital
7
3.
Equations
Simplified
Model I (D-CLoad Flow)
(3) and (4) can be greatly
simplified
if we make the
following assumptions:
a) R is small compared to X. Transmission lines are
usually designed so that
R
IXI/10.Since real power
losses are related to R, utilities are interested in
having small line resistances.
b)
The bus voltage magnitudes V i and Vj are almost equal.
This is usually true in practice, since too large a
variation in bus voltage magnitudes can make for
damaged equipment, unreliable
customers.
operation and angry
V i is usually equal to Vj to within plus
or minus 5%.
c)
The difference in voltage angles (6i - 6J) is small.
It is a fact that (6i
- 6J) seldom exceeds 150,
for stability reasons.
In this range of values:
sin(6i
- 6) - (6i
cos (6i - 6j)
Using these
Pij
6j)
1
approximations
X
mainly
(i
6j)
(j
- ~i)
we obtain
for Pij and Pji:
V2
Pji
X
( )
We can obtain a similar expression for PL by using the
approximations in equation (5); we get:
R
PL
2
2
v
ij
If V is expressed in per-unit, and if we assume that
Vi
i
=V.
J
1 PU:
1
ij
PL
(6i- 6J)
X
= R ij
(9)
ij
(10)
We shall from here on refer to the linearized model described by
equation
9 as the simplified
model
I.
It is often
called
a D-C load
flow solution.
3.1
Analogy between the simplified model and a resistive network
Let us consider a resistive network where the only active elements
are current sources connected to ground, as in figure 3.
It is easily
seen that equation (9) expresses the relationship between the node
voltages and the currents in the resistors if we assume that the real
power flows PiJ correspond to the currents Iij, that the phase angles
6.icorrespond to the node voltages vi, and that the line reactances
Xij correspond to the resistances rij.
Since the sum of the currents
at a node must be zero, the intensity of the current sources Ii must be
equal
to the load or generation
at bus i.
Similarly,
equation
(10) can
be interpreted as a correspondence between the line losses in the power
system and the heat dissipation in the resistive network.
Notice,
9
v2=62
v =61
3 =6 3
11
Figure
3
however, that the line losses are proportional to Ri,
analogy also existswith the voltage drop in resistor
and not Xij. An
(ij) and the
reactive power in line (ij). These conclusions are summarized in table I.
10
Table
I
A Poor Man's View of Power System
A power system can be modeled by a resistive network if the
following equivalences are made:
Power System
Transmission line with
Resistive Network
Resistor
Xij
reactance Xij
Generator or load of
intensity P
Current source of intensity
Ii
Pi
Bus voltage angle 6.
Node voltage v i =
Real power losses in line ij
Proportional to heat losses
in resistor (ij)
Reactive power in line (ij)
Proportional to voltage drop
in resistor (ij)
i
11
Table II
Table III
Complete System
Line No.
Linear Model
Line No. 16 Out of Service
Load Flow
Linear Model
Load Flow
1
79.6
80.5
107.9
111.3
2
103.9
106.9
132.2
140.9
3
127.2
124.1
197.8
195.7
4
93.5
90.7
164.0
156.2
5
15.2
16.8
60.12
56.9
6
20.4
20.?
41.4
39.4
7
26.2
26.0
50.2
48.0
8
52.65
73.7
69.9
9
51.55
51.4
52.5
107.4
107.0
10
174.9
172.6
275.6
267.6
11
108.6
106.4
224.1
210.2
12
27.45
27.8
-8.1
-16.7
13
?6.45
27.0
-8.1
-15.7
14
33.35
33.4
-87.4
-85.8
15
33.35
33.6
-87.2
-85.3
0
0
16
241.5
239.7
17
252.4
250.8
11.1
10.1
18
185.8
183.8
286.4
281.5
19
267.8
265.8
26.5
24.3
20
201.15
198.4
301.8
296.1
-149.7
-152.4
481.6
482.2
279.6
277.0
21
-191.4
22
439.8
23
273.8
-193.3
434.9
233.9
24
135.8
134.0
177.6
175 .6
25
84.8
84.0
126.6
125.7
Line losses:
40.3 W
Line losses - 22.5 W3r
12
2.2
Matrix solution of the Simplified Model I
Let PZ be the vector of line flows, and Pb be the vector of bus
injections.
Then Pp can be computed from P
P
-/
T
ArA
=Y
`e-r rr
Y
-1
Pb
-r
(11)
Yp :
diagonal
A
reduced bus incidence matrix
where:
:
as follows
matrix of line admittances
See reference (11) for further details.
3.3
Accuracy of Simplified Model I
Simplified model I usually gives, for the real power flows, values
accurate to
5%.
For example, a comparison of lines flows computed
using a load flow and the linearized model for a part of the New England
power system is given in reference (2).
methods is excellent.
The agreement between the two
When large errors occur, it is for lines connected
to the swing bus, whose generation is adjusted by the load flow to compensate for
3.4
line losses.
Uses of Simplified Model I in the literature
Simplified Model I is used often and for many purposes.
Here are a
few articles making use of this model:
tIt is repeated here (tables II and III) for easier reference. The
network is the same as the one used to illustrate Simplified Model II.
13
N. V. Awanatidis
"The use of objective functions in real power
Dispatching."
V. Rosing
(IEEE Winter Power Meeting, 1971)
The authors discuss economicdispatching using
equation
(11) as a constraint
"PowerSystemStatic-State
F. C. Schweppe
D. D. Rom
Approximate model."
I,
in a linear
estimation.
program.
Part II:
Vol. PAS 89, No. 1,
January 1970.
The authors use Simplified Model I in discussing
filtering techniques for state estimation in power
systems.
J. C. Kaltenbach
"Optimal corrective
L. P. HadJu
Security."
A. Thanikachalam
J. R. Tudor
rescheduling for Power System
IEEE PAS 90, March-April 1971.
"Optimal rescheduling on Power for System Reliability."
IEEE PAS-90, Sept.-nct. 1971.
The authors discuss the rescheduling problem using,
again, equation (11) as a constraint
L. P. Hadju
D. W. Bree
in a linear
program.
"On line monitoring of Power System Security."
IEEE PAS, Feb. 1963.
A. W. Brooks
G. A. MacArthur
"Transmission Limitations Computedby Superposition."
AIME transactions, December 1961.
H. . imer
"Techniques and applications of Security Calculations
Applied to Dispatching Computers."
PSCC Proceed-
ings, Rome, June 1969.
D. Crevier
"Steady State Contingency Evaluation of Power
Systems."
M.S. thesis,
.I.T., 1971.
These papers all apply Simplified Model I to the problem
of contingency evaluation of line faults.
3.5
Disadvantages of Simplified Model I
For certain types of studies, Simplified Model I still requires
too much computation.
inversion.
the
For example, equation (11) requires a matrix
In metwork expansion studies, where it is desired to analyze
effects of a multitude of network configurations,
cumbersome to modifv the inverse
bus admittance
matrix
everytime. This brings us to the next model.
it is extremely
ArT Y1 Ar 1
15
4.
Model 2 (Transshipment Model)
Simplified
we wantedto find the current distribution
Suppose
that minimizes the heat losses in the resistors,
current sources at
That
nodes.
the
is,
in Figure3
consistent with the
we want to solve the following
mathematical program:
MP I
z =
Min
P
P
subject to
that
Notice
Xi
ij
Xij
at every node i
P.
1
ij
into
we are deliberately not taking
voltage law in our constraint set.
account Kirchhoff's
Instead, we rely on the minimization
specify the
of the objective function to completely
currents.
We then
verify the following interesting result:
"The currents Pij that minimize the objective function
above are precisely those that we would obtain by
solving for the currents using both Kirchhoff's
voltage and current laws."
In other words, the natural current distribution is the one that produces the minimum amount of heat losses.
The mathematical program above can be written as:
Proof:
Min
T yl
z
T
A
-r
subject to
P
(12)
p
= P
- b
(13)
o
We must prove that:
-1
1)
P
0
= Y
AP
T
Y
Ar]
b
O
is a solution of (13).
16
2)
For ay
P,
such that
(13) holds,
P
1
Pe
YTT P
X1
pi
y0 p
We first prove (1):
T
=A
r-o
-.
0
Clearly
A TP
So
P£
Y ArAY
is a solution
-1
Al
-b
=b0 '
0
of (13).
0
To prove (2):
Lemma:
Le-t
Pb
b=
[A TPby
-P!b
lb
=
Then P
10
i
Proof of (2):
-1
A
Y
Ar
%P =Y AP
I
o
r
Consider now an, other solution
P
such that
'1i
:TTP =Pb
T
AT P
-r
Then
-11
0
Pb ATP
0
= (A Pb )T
(yl
,
b
AT
:P
_
T(ArTP
1 b
1~~~=
)T
-I0
I~~~~~~
Also
p
T (ATP
1
T
'e
)
= (E
(
P T Y -1 Pl
We therefore have
0o
which implies
T
=
1
T Y--1 P
-0
-PT) - P =0.
-l1 !e Pro
Completing the square we get:
--o
-1
'
T
A
i7
)P
Y-1)
0o
1-
17
'(oT
.... -Ptl
'o-"-to
TI-o
T-'J'P-"o
'
T -I .1
=P T0>o0
(P TYp
-
2T-2Y-lp + _TY -lPS
T-l T
"%-=P
MP
2:
T-
z
M'in
subject
to
(
TJT
Xil=Pij
-/
T
-P) YP
-1
)T -1
=o
-
I
Pij = Pi
at every
node i.
This program is identical to the previous one except that we have
replaced the quadratic cost function by an absolute value cost function,
as illustrated in Figure 4.
Intuitively we should expect some correlation between the solutions
to MPI and MP2 because of the similarity of the cost functions.
known in operations research as the transshiment
roblem,
MP2 is
because it
can also express the problem of minimizing the transportation cost of
goods that have to be shipped across
a given transportation network.
It is, in fact, pretty much the problem we would have to solve if we
_
~P
-.
.
i~
-,
~~~P.,
P4 4
1
1
1
iLJ
Pij
Absolute value cost function
Quadratic cost function
Figure 4
wanted to study the transportation cost of energy in terms of fuel
(coal, gas, etc...) rather than electricity.
4.1
Peculiarities of the transshinment problem and solution
P2 is not an ordinary linear program.
Actually it is one of the
simplest cases in the large class of transportation
problems.
The fact
that the sum of the power flows at all nodes must be zero gives the
problem
a particular structure that makes it amenable to a very simple
and efficient solution. We give here a description of the algorithm
that is by no means intended to be a rigorous mathematical
For a better treatment, the reader is referred to Danzig
Wagner
description.
(17) or
(18).
Consider the network in Figure 6.
Flows of 1, 2, and 3 units are
injected into buses 1, 2, and 3 respectively.
A flow of 1+2+3 = 6 units
19
Demand
6
6
6
6
Figure
5
R. =1
Figure 6
is coming out of node 4.
Suppose we want to find the flow distribution
in the lines that will minimize
E
i
R.ij1Pij . The problem can be put
in the form of a tableau (see Figure 5).
A row and a column are assigned to each node.
costs of sending one unit of flow from i to J.
The Cij 's are the
In this example, the C
matrix is full because every node of the graph is connected to every
20
In most cases this is not so, and the elements of
other by one line.
the tableau corresponding to non-existent lines are left blank. The
Pij 's are the line flows we are seeking to determine.
columntotals
Pij and ~
(i
The row and
Pi) must be respectively equal to
the supply and demand at the corresponding nodes plus a constant pre"Self-flows", as Pll' PP' 2
venting the flows from becoming negative.
are fictitious.
A.2
Analytical formulation of the problem
The above tableau is equivalent to the following linear program:
m
Minimize "5
(1)
m
Y ci P
i=1 j= ij
subject to: (2)
(3)
Z
Pij =Si
for i
1,2,...m
(supply)
E
Pij = D.
for J
l,2,...m
(demand)
j=l
i=j.
J
J
for all i and
(4) Pij Z 0
m
m
(5)
(m number of nodes)
ij
2
i=l
S. = 2
j
D.
(total supply = total demand).
j=1i
It can be shown that one of the 2m constraint equations (1) and (2)
is redundant.
Any solution of the program will therefore contain only
2m-1 nonzero Pij 's.
(It is a general property of linear programs
that the optimal solution, or any intermediate solution in the computation thereof, contain as many nonzero variables as there are restrictions, at most.)
21
Algorithm
Step 1: Start with a basic feasible solution; i.e. a set of
2m-1 nonzero Pij
's satisfying
the row and column
summation constraints.
Step 2:
Check whether the solution is improved by introducing
a nonbasic variable (one whose value is zero in the
current solution).
If so, go to step 3.
Otherwise,
stop.
Step 3: Determine which variable leaves the basis whenthe
variable selected in step 2 enters.
Step : Adjust the flows of the other basic lines.
Return to
step 2.
Example
Step 1:
Suppose we start with the following basic feasible solution in the above problem (left-hand tableau).
t(
fr
"
1
1
2
ate
3
fr~ .1
/4
1
2
3
1
0
51
2
6
3
2
2
0
61
4
4
6
6
6
6
5
0
0
1
8
2
9
3
3
0
4
4 --
o
1
0
/.
4
-5
-8
0_ 4
2
8
-1. 2 -4 1 061
-4
Figure 7
3
1
7
6
3
2
616
-131
-8
6o
°
10
-121
-6
0
0
6
O
22
Step 2:
The right-hand tableau contains what could be called the
negative values of the partial derivatives of the objective function with respect to nonbasic (zero) flows.
In
this case it can be seen that introducing one unit of
either P21 or P24 would improve the objective function by
three units.
Any other modification would produce an
increase in the objective function.
The right-hand
tableau can be deduced easily from the left one.
Step 3 + 4:
We would introduce, say P2 1 into the basic solution, and
modify the other flows so as to keep row and column
totals constant.
We would in the process eliminate P23.
Returning to step 2, the right-hand tableau would show
this new solution to be optimal.
Remarks:
1)
The solution to this problem can be reached very easily by
hand in a few minutes.
The only arithmetic operations per-
formed are additions and subtractions.
It is the author's
experience that much larger problems (say 20 nodes) can also
be solved by hand rather easily.
Solving Kirchhoff's equa-
tions for a network of this size is a horrendous task, if the
inverse bus admittance matrix is not readily available.
2)
For these reasons, it is felt by the author that the saving
in computer time realized by using the transshipment problem
rather than Kirchhoff's laws can be of one or two orders of
magnitudes.
This is especially true when the problems have
to be solved from scratch (i.e. when the admittance matrix B
23
has to be inverted), or when it is desired to study the effects
of variations in network geometry (in which case the B -1 matrix
must undergo onerous modifications.
See (9).
4.3 Accuracy of the transshipment algorithm
Here
is a
comparison of the solutions to the network of Figure
5,
as obtained by the algorithm, and by using Kirchhoff's law:
Flow
Flow,
TSSP
Flow,
Kirchhoff
3
I
I
h
N W
F
IQ
I
2
1
2
0.648
2
0
O.593
3
0
0.611
4
3
2.30
5
0
1.968
6
3
1.84
El
1
El1
I
1
6
4
Line no.
d) Kirchhoff
0 TSSP
Table
II
Figure 8
Furthermore, here are similar results bearing on an 18 bus,
system which is a part of the New England power system.
power flows are the results of a load flow study.
display graphically the results of tables 2 and 3.
5 line
The "exact"
Figures 8 and 9
In figure 9, the
lines have been ordered according to the intensity of the flows, so
as to render the correlation between the two curves more apparent.
24
Table
System Data
Admittance
Bus
No.
load ()
Generation ()
Line
No.
5
5.77
13.80
9.43
28.40
111.20
6
17.35
7
19
18.05
188.80
18.12
27.10
86.30
8.29
8.00
12.90
12.90
65.00
62.90
63.30
34.70
20
34.70
21
72.40
52.60
60.50
28.50
12.22
1
(SWIm BUS)
2
?3 .3
0.0
2
3
0.0
32.8
0.0
508.0
3
0.0
0.0
0.0
0.0
0.0
0.0
4
40.0
10
11
9.9
0.0
0.0
0.0
0.0
0.0
632.0
15
200.0
100.0
50.0
0.0
0.0
0.0
16
4
5
10
29.4
30.3
42.5
171.0
211.8
11
9.9
12
14.4
14.4
6
7
8
9
13
14
15
16
17
18
(p.u.)_
8
9
12
13
14
17
18
22
23
24
25
25
iH
~i
c)
a
C12,
D
qDj
-
xc
ri
P
+
Ur
H4
W
U)
:-
26
Line
Flow,IvIA,
TSSP
1
2
3
4
5
6
7
8
9
10
11
12,13
0
24.3
33.8
0
31.4
0
0
32.3
0
76.8
31.4
0
Flow,MVA,
Ioad
Line
Flow
80.5
106.9
124.1
90.7
16.8
20.2
26.2
51.4
52.5
172.6
106.4
54.8
14,15
16
17
18
19
20
21
22
23
24
25
Flow,MVA,
Flow,YMVA,
TSSP
Load Flow
204.4
379.2
390.1
87.6
405.5
103.0
58.6
573.4
371.4
269.4
218.4
67.0
239.7
250.8
183.8
265.8
198.4
193.3
434.9
233.9
134.0
84.0
Table 3
45
40(
35(
30(
25(
20(
15(
10(
loc
5C
2
Load Flow,
Lines
TSSP(
Figure 9
27
It can be readily ascertained from these graphs that there is
indeed a correlation between the results of the transshipment algorithm
and the load flow or Kirchhoff's laws.
The reader may have observed,
though, that in general the small values of flows, computed by the
transshipment algorithm are too small, and the large values are too
large.
For example, all transshipment flow values above the average
value of the flows in Figs. 8 and 9 are above the load flow curve.
All transshipment flow values below the average value of the flow are
below the load flow curve, except one.
This fact can be explained as
follows:
a)
Remember:
the transshipment curves in Figs. 8 and 9 are the
solution to MP1.
The load flow and Kirchhoff's laws curves
are associated with MP?.
Both objective functions penalize
flows in high impedance lines.
However the quadratic cost
function in MP1 also penalizes large values of flow.
does not.
P2
In MP2, the flow will tend to concentrate more in
low impedance lines, than in MP1.
b)
Since more flow circulates in low impedance lines in MP2, more
of the load is met through these lines and less power is left
to be carried by the high impedance lines.
Actually, it can be
proved that the optimal solution to MP2 will assign a flow of
value zero to a number of lines equal to the number of lines in
the network minus the number of buses; the lines carrying a flow
are seen to form a tree in the network (except in degenerate
cases).
4.L
Smoothing out procedure
Based on these observations, Figs. 10 and 11 have been drawn,
using the following Smoothing out procedure.
If a flow computed by
the TSSP algorithm is smaller than the average values of all flows
computed in this fashion, one half of this average value was added to
it.
Otherwise one half of the average value was subtracted from it.
It can be seen that the corrected curves follow much better the "exact"
curves.
In the case of Fig. 11, the average error is 35 MVA, or 20%
of the average value of the flows.
Remarks:
1)
The flows obtained from the transshipment algorithm obey
Kirchhoff's node law.
2)
The corrected flows do not.
We do not consider here the direction of flow.
In a few
instances, for small flows, the TSSP algorithm gives flows
in the wrong direction.
assuming the
Furthermore, we have no way of
irection in which to correct the flows to
which the value zero was assigned by the algorithm.
However,
if the results of the algorithm are used to compute losses
or to detect overloads, the direction of the flows is
unimportant.
3)
Even though the magnitude of the errors is small in Fig. 11,
the percentage error can be exceedingly large for small flows.
This could perhaps be corrected by using a better smoothing
procedure:
for example we could try to add less to the small
flows and subtract more from the large ones.
Extensive statisti-
cal studies would be needed to establish the optimal smoothing
29
..
Line
2
3
1
6
5
4
Figure 10
450
400
350
300
250
200
150
100
50
A rf
0
5
6
7
8
9
12 14 1
25
4
11
2
3 24
13 15
Figure
11
10 18 21
20
23 16 17
19 22
30
In any case,
procedure.
errors on small flows are not too important
since:
a) They areon the safe side:
flows are
the indicated
largerthanthe actual ones.
Even though
b)
larger than the actual
ones, the
flows are still comparatively small.
larger
than
the overload
indicated
They may not be
limits of the corresponding
lines.
4.5
Computing Line Losses
If line losses are computed
using the
results of the
non-smoothed
transshipment problem and equation (10) which can be written as:
PI
PL = 2
L
R P2
=
K
K
(14)
KK
it can be expected that the computed losses will be much too large;
this is due to the fact that the losses depend upon the square of the
line flows.
Small line flows contribute therefore proportionally less
to losses than large flows, and the large flows are too large in the
transshipment model solution.
If
the
smoothed out solution is used this
effect disappears and the estimated losses should be pretty close to
reality.
4.6
Uses of the transshipment algorithm in the literature
The transshipment model of a power system is mostly used for
planning purposes.
Garver, (6) Garver.
See for example (8) Marks Jirka, (7) Platts, Sigley,
The following justification for using the trans-
31
shipment model is given in Marks, Jirka:
"...Although there certainly ... exists a combination of
fixed and variable costs with economies of scale involved,
a linear representation with a constant price of transportation per unit of commodity transported is reasonably
valid in many cases.
This is also applicable for electric
power transmission, where transmission cost are largely
amortized fixed charge costs plus additional costs of
maintenance and power losses, all of which do not exhibit
strong economies of scale."
We note that planning studies take into account more than power
losses:
line construction
and maintenance costs are also considered.
It is the author's opinion that, if the flows are not smoothed out,
this approach can be misleading for the following reasons:
1)
In the unsmoothed solution, no power will flow in a certain
number of lines, and these lines will make no contributions
to the linear objective function (
Z RKPK).
However amor-
K
tization and maintenance costs are still associated with these
lines, and the linear objective function is not realistic.
2)
Power losses are quadratic, not linear.
Once more the linear
objective function is not realistic.
The conclusion to draw is that, even if a transshipment-type
algorithm can give a pretty accurate idea of the line flows for a
particular network configuration, it does not follow that the value of
32
the optimal linear cost function associated with this configuration
has any relation
to the cost of buildingand operatingthis configura-
tion.
A more accurate cost can be obtained by using the smoothed solution
and computing the value of a two component cost function, as follows:
a)
A quadratic component associated with line losses,
b)
A linear component associated with maintenance and
amortization.
If a choice is to be made between
be based
upon
several
configurations,
it should
the value of the two-component cost function, rather than
the value of the cost function used to carry out the minimization.
Another important point:
if the flows computed by the TSSP algorithm
are to have any relation with reality, the parameters of the objective
function must be associated with the impedances of the lines, and not
with the cost of building the lines, as in most economic studies.
For a
given voltage level, a low impedance line will cost more than a high
impedance line.
33
5.
Simplified Model 3 (Ford-Fulkerson Model)
c3
ci: capability
of line i
Figure 12
This model is an extremely simplified version of the problem.
POnlyline capabilities are considered.
admittances are taken into account.
No transmission cost or line
Basically, the aim of a Ford-
Fulkerson study of a network is as follows:
for a given load-generation
configuration, see if there exist a solution to the transshipment problem (the capabilities of the lines could be too small for a solution to
exist).
If such a solution exists, the algorithm will provide a feasible
solution, which can be used as a starting solution by the transshipment
algorithm.
If no solution exists, the Ford-Fulkerson algorithm can discover
the "bottleneck":
it can be shown that the maximum amount of power that
can flow from generation to load is equal to the capacity of the
"minimal cut" from generation to load.
(Ford-Fulkerson, "Flows in
34
Networks").
This minimal cut is a set
of lines, and if the capability
of these lines is increased, the capability of the whole network is
increased.
Note:
Obviously, if there exists no solution to the Ford-Fulker-
son algorithm, it is impossible to meet the load without overloading a
The reciprocal is not true:
line.
the existence of a solution to the
Ford-Fulkerson algorithm does not imply that a load-flow solution of
the same problem would not result in a line overload.
This is so
because the load-flow solution has to meet more constraints (e.g.
Kirchhoff's voltage law) than the Ford-Fulkerson solution.
Advantages and uses of the Ford-Fulkerson algorithm
5.1
The algorithm has the advantage of being computationally very fast
See
and efficient, even much more so than the transshipment algorithm.
(16) for more details.
It is mostly used in the first stages of network
design, for example to test a proposed design against line outages,
varying load configurations, etc....
Studies using this algorithm were done in 1968 by Electricite de
France to
decide upon important additions to be made to the French power
grid.
5.2
Computing line losses
Line losses can be computed approximately using equation (10),
repeated here for easy reference:
PL
= Rij PIJ
Lij = Rij PiJ
(10)
35
Rij
represents the resistance in a line going from i to
Pij
is the real power flow in this line computed by any
of the previous algorithms.
This formula gives very accurate results (
with Simplified Model 1.
5%) for flows computed
See (9) for example.
Judging by the accuracy of the flow values, an accuracy of 10 to 20%
should be expected with Simplified Model II on the sum of the losses in
all lines, if the smoothing procedure is used.
It is the author's opinion that formula (1) should be used with
great caution with flows computed by the Ford-Fulkerson algorithm,
sincenothing indicates that these flows have anything to do with the
physical flows that would be observed.
36
6.
Conclusion
We have discussed different models of power systems, and simplified
algorithms for computing line flows and losses.
As could be expected
the accuracy of the results decreases with the simplifications involved
and the rapidity of the algorithms.
Generally speaking, the Ford-
Fulkerson algorithm can give a rough idea of the capabilities of a network, by detecting certain overloads.
properly
mate
chosen,
If the objective function is
the result of the transshipment algorithm will approxi-
the actual electrical flows that would be observed in the network.
The accuracy will be increased if a smoothing out procedure is used.
Simplified Model I gives very accurate results, and quantities such as
reactive power flows can be deduced from the results.
It is the author's belief that, if used knowledgeably, the transshipment and
ord-Tulkerson algorithms can result in considerable
savings in computer time for studies where accuracy is not a primal
concern.
37
Bibliograoph
Papers
(1) Arvanitidis, N. V., Rosing, J., "The use of objective functions
in real
power dispatching."
IEEE Winter Power Meeting,
Nw York,
Jan. 31 - Feb. 5, 1971.
(2) Baleriaux, H., et al., "Optimal investment policy for a growing
electrical network by a sequential decision method." CIGRE International Conference on Large High Tension Electric Systems 1970 Session.
(3)
Auge, J. et al., "Probabilistic study of a transmission system
interconnection." CIGRE - International Conference on Large High
Tension Electric
(4) Henult,
Systems - 1970Session.
P. et al.,
"Power System Long-TermPlanning in the
Presence of Uncertainty." IEPE, Pas-89,
(5)
(6)
Kaltenback, J.
1970.
Mathematical Optimization Technique
for the Expansion of Electric
Pas-98, Jan. 1970.
Power Transmission Systems."
Garver,
Network Estimation
L. L., "Transission
Programming."
(7)
C. et al., "
Jan.
Platts,
IEEE, Ps-89,
J. E. et al,
Planning."
1972.
IEEE,
Using Linear
Sept. - Oct. 1970.
"A Method for
liorizon-Year Transmission
IEEF Winter Power Meeting, New York, Jan. 30 - Feb. 4,
(8)
Mark, D. H. and Jirka, G. H., "Environmental Screening Model for
the Location
f Power Generating Facilities." Presented at the
Thirty-Ninth National Meeting,
Operations Research Society of
America, Dallas, Texas, May 7, 1971.
(9)
real
Boughman, M. and Schweppe, F. C., "Contingency evaluation:
flows from a linear model." M.I.T. E.E. Dept., unpublished.
(10)
Schweppe, F. C., Rom, D. B., "Power System Static-State Estimation,
Part II, Approximate Model." IEEE, PAS-89, Jan. 1970.
(11)
Crevier, D., "Steady State Contingency Evaluation of Power Systems."
M.I.T., E.E. Department, M.S. Thesis, Nov. 1971.
(12)
Kaltenbach, J. C., HadJu, L. P., "Optimal Rescheduling on Power
for Systems Security." IEEE, PAS-90, March 1971.
(13)
Thanikacnalam, A., Tudor, J. R., "Optimal corrective
rescheduling
for Power System Security."
IEEE, PAS-90, March - April 1971.
(14)
Hadju, L. P. et al.,
TEEE, PAS, Feb.
(15)
"O n line monitoring
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1963.
Limmer, H. D., "Techniques and applications
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Calculations
applied to Dispatching Computers." PSCC, Proceedings, Rome, June
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(16)
Ford and Fulkerson, "Flows in
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Dantzig, G. B., "Linear Programming and Extensions."
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(18)
Wagner, H. M., "Principles of Operations Research."
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